Bayesian Example

Question

Ex.

suppose that $x=2$ denotes the number of successes in $n=5$ independent trials with probability $θ$ of success, that is $x$ has a binomial distribution with the parameters $n=5$ and $ θ$. assuming that the parameter $θ $ has a uniform(zero,1) prior density:

(a)find posterior density function, and hence the posterior mean of $θ$.

(b)find a two tailed $90$% credibility interval for θ in terms of the inverse posterior distribution function.

(c) test the hypothesis$ H:θ=0.3 vs A:θ≠0.3 $, assuming that the prior probabilities of H and A are equal.

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Suggested answer

(a) Posterior $ 1/β(3,4) * θ^2 * (1-θ)^3 $, and mean $3/7$ .

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(b) Two tailed 90% credibility interval calculate it from $x ̅±z_(α/2) *σ,x ̅=3/7 , z_(α⁄2) =1.65 , σ=√(3/98) $ .

(but i do not understand meaning of "in terms of the inverse posterior distribution function")

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(c) Calculate from
$ p(H|x)/p(A|x)=p(H)/P(A) * P(x|H,θ)/∫P(x│A,θ) P(θ) dθ $

Where $P(H)=P(A)=.5$ ,

$P(x|H,θ)= 10 * (.3)^2 * (.7)^3 =0.3087 , ∫P(x│A,θ)* P(θ) dθ =(10/√2π)∫θ^2 (1-θ)^3 exp(-.5 x^2) dθ ,x=2$.

and finally result $p(H|x)/p(A|x)=.00129 <1 $, then reject $A:θ≠0.3$

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Please tell me about my answers true or false and in (b) what mean about terms of the inverse?

Any help would be appreciated

Thank you so much!

Answer 1

The prior distribution is $UNIF(0,1) \equiv BETA(1,1)$ with density function $p(\theta) = \theta^{\alpha_0 - 1}(1 - \theta)^{\beta_0 - 1},$ where $0 \le \theta \le 1$ and $\alpha_0 = \beta_0 = 1.$

The likelihood function is proportional to $\theta^2(1-\theta)^{n-2},$ where $n = 5.$

The posterior distribution is found as $$Posterior \propto Prior \times Likelihood.$$

So your posterior distribution is (proportional to) $$p(\theta|x) = \theta^{3 - 1} (1 - \theta)^{4 - 1},$$

which is $Beta(3, 4).$

Therefore, a 90% posterior probability interval can be found by cutting 5% from each tail of $Beta(3,4).$

In R, this can be evaluated as $(.153, .729)$:

 qbeta(c(.05,.95), 3, 4)
 ## 0.1531611 0.7286616

The posterior is BETA(3,4), and the quantile function of a beta distribution (inverse CDF) is called qbeta in R.

Note: You seem to be using a normal approximation to this beta distribution, which is not something I would do.

I will leave the test of hypothesis to you.